We investigate the joint moments of the
power of the characteristic polynomial of random unitary matrices with the
power of the derivative of this same polynomial. We prove that for a fixed
, the moments are given
by rational functions of
up to a well-known factor that already arises when
We fully describe the denominator in those rational functions (this had already
been done by Hughes experimentally), and define the numerators through various
formulas, mostly sums over partitions.
We also use this to formulate conjectures on joint moments of the zeta function
and its derivatives, or even the same questions for the Hardy function, if we use a
“real” version of characteristic polynomials.
Our methods should easily be applied to other similar problems, for instance with
higher derivatives of characteristic polynomials.
More data and computer programs are available as expanded content.